Distances, diameters and verisimilitude of theories

The verisimilitude problem, that is what it means when we affirm that a (false) theory T is closer to the truth than a (false) theory T', plays a crucial role in Popper's philosophy of science. Now, as observed in [6] by Miller, "If theories can be close to or distant from the truth, then presumably they can be close to or distant from one another". This has led several authors to define in the class g of theories under consideration by the scientific community, a distance-like function (see for example [4, 6, 7, 8]) and subsequently, to define the verisimilitude as a function of the distance from the truth. In this paper we propose an approach to this question via the concept of "pointless pseudometric space" and "pointless metric space" (cf. [111 and [1, 2, 3]). Namely, a "distance" 6 :g x 3---->[0, oe) and a "diameter" I I: ~--->[0, oo] are defined in J and, if V denotes the set of true sentences, then the verisimilitude of a theory T should be defined as a decreasing function of 6(T, V) and ]T]. The so obtained definition avoids the well known Miller-Tichy's results about the incomparability of false theories.